I gave a challenge question at the end of class a week or so ago. Here I will give the solution and show that it gives us something rather strange and surprisingly useful.
I wrote down the following, and asked you to prove it:
For 
And the number of ways of pulling no objects out of a bag is 1, because you just don’t pull anything out.…
Out of the blue I wrote down a rather confusing mass of indices and summations on the board a few days ago. Writing this down at the last minute was perhaps a bad idea, but I wanted to show what the general form for expanding a fraction into partial fractions was. Here I’m just motivating it a little more. It’s not something that you will need to use, but it’s often good to write things down in as general a form as you can.
Let’s say that we have an expression of the form:
Where P(x) is some polynomial of degree less than 3 (because the denominator is degree 3). We can write this as:
To find A, B and C, you cross-multiply, and then match coefficients of powers of x with those in P(x). If you have an irreducible quadratic in the denominator you will have terms of the form:
in your partial fraction, and of course if it’s an irreducible quadratic to an integer power greater than one, you will have multiple terms, just as you do for the (x-2) expression in the example above.…
I was so excited the first time I read Ian Stewart’s book entitled “The 17 equations that changed the world“. The book is written in simple and easy to understand language with interesting practical examples for applications. I immediately wrote to Ian Stewart requesting if I could reproduce his work in form of posters in both English and French to be used at AIMS-IMAGINARY in Senegal in 2015 (see here). I remember his only reply was “please proceed but I won’t be able to attend since I have prior committments”. Business Insider published a list of these equations emphasing further how intuitive they are (see here). I do strongly believe every school in the world be it elementary, college, secondary, technical, university, you name it, should have these posted up or painted on the walls of their science departments/offices, classrooms, laboratories etc; in all langauges applicable.…
I came across the following YouTube video a while back which uses a strange trick to accurately approximate square roots. I suggest watching at least the first minute where the presenter explains how it’s done:
Let’s do an example. Approximate to 2 decimal places:
First, YouTube tells us to find the nearest perfect square that’s less than 40, that’s 36, and take the root, giving us 6. So our answer is, obviously, 6 point something. That something is a fraction, where the numerator is the difference between 40 and 36, and the denominator is 2 times 6. So we have:
So what’s the actual answer to 2 decimals? It’s 6.32. That’s what I like to call: pretty darn close. Naturally, there are a few catches to this technique: you need to know your perfect squares, you need to know your fractions, and things tend to get hard with larger numbers.
But still, I thought this was surprisingly effective for such a simple piece of arcane trickery.…
Hello Internet. In this blog post we are going to learn some type theory, which is a field that lies in the intersection of mathematics and computer science. We are eventually going to learn the variant called Martin-Loef Type Theory, or MLTT for short. To motivate it, we are going to try answering the question, “What is a valid program?”
Not every program makes sense, for example what would a program consisting of the single expression x + 2 mean? If we don’t know what x is, then we don’t know what the meaning of that expression is. More importantly, a program can combine expressions that are incompatible with each other. For example, if + is the usual operation that adds two numbers then an expression like "abc" + 5 can’t be given a meaning. To avoid such meaningless programs, we should talk about the different kinds of data and the valid operations on these pieces of data.…
The following has been a rather interesting journey – from a test question which seemed fine, to a subtlety which seemed easy, to a discussion with a number of different mathematicians about the nature of distributions, measure theory and regularisation. I will try and make it as clear as possible in the post below. Note, as mentioned in the comments, we have actually only found the solution to this problem for a constrained range of x, and not x 
In a recent class test, there was a question, written by me, which was not quite the question that I wanted to ask. It turns out that it does have an answer, but it’s not an answer that can yet be found by the means at the class’s disposal.…
Ever since Euclid first proved that there are infinitely many prime numbers, mathematicians have found ever more creative ways to prove the same result, and also various stronger theorems that imply it. Dirichlet’s Theorem, for example, states that if
and that the number of prime numbers less than 
To do so, we will look at the convergence of two different sums: that of the reciprocals of the primes with a prime number of digits, and that of the reciprocals of the natural numbers which do not contain your phone number amongst their digits.…
By: Jan Wuzyk
In this article I am going to show that nowhere differentiable functions do in fact exists and give a few examples, some of which are relatively modern. But first I’m going to try to answer a question that is, in my opinion, too rarely discussed in mathematics classes, ”Why do we care?”.
To answer this question we have to look into the history of mathematics. In 1821 Augustin-Louis Cauchy published his seminal book, Cours d’Analyse, this is generally recognized as the first serious attempt to put calculus on a rigours footing[Com][1] , mainly through introducing rigorous definitions of limits,continuity and differentiability among others, and the definitions that go with them2. This was also time the integral was defined as an area instead of simply as the antiderivative.
It should be noted that Cauchy by no means closed the issue of rigour in analysis but he provided a starting point.…
How would you go about finding the value of 
or, in other words, that x, which, when squared gives 3. We have to be a little careful here because we know that there will actually be two numbers which satisfy this (one positive, one negative), and we are interested in the positive one only.
So, we try some values, but we don’t do it randomly, we can see that because 
We’ll start with the “classic” example. Consider the data plotted in figure 1. Each data point has 3 “properties”: an 
Figure 1: A scatter of data with three properties: an x_1 coordinate, an x_2 coordinate and a colour.
Suppose for each data point we generate a representation of the data point ![\phi(x)=[x_1, x_2, x_1x_2]](http://s0.wp.com/latex.php?latex=%5Cphi%28x%29%3D%5Bx_1%2C+x_2%2C+x_1x_2%5D+&bg=ffffff&fg=000&s=0 "\phi(x)=[x_1, x_2, x_1x_2]")
